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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001280
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001330
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 43%●distinct values known / distinct values provided: 25%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 43%●distinct values known / distinct values provided: 25%
Values
[{1,2},{3}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[{1,3},{2}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[{2,3},{1}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[{1},{2,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1},{2,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1},{3,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2},{1,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2},{1,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3},{1,2},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{4},{1,2},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3},{1,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{4},{1,3},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2},{3,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3},{2,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{4},{2,3},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,2},{3},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,2},{4},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,3},{2},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,4},{2},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,3},{4},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,4},{3},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,3},{1},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,4},{1},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3,4},{1},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,3},{4},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,4},{3},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3,4},{2},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,2},{3,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,3},{2,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,4},{2,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,3},{1,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,4},{1,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{3,4},{1,2}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,2,3},{4}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1,2,4},{3}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1,3,4},{2}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{2,3,4},{1}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1},{2},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{2},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{2},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{3},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{3},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{4},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{5},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{4},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{5},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{3},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{4},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{5},{3,4},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2},{1},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2},{1},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2},{1},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3},{1},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3},{1},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4},{1},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{5},{1},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4},{1},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{5},{1},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3},{1},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4},{1},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{2,3},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2,4},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2,5},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3,4},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3,5},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{4,5},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1,3},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1,4},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1,5},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{1,2},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{1,2},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{1,2},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{1,4},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{1,5},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{1,3},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{1,3},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{1,5},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{1,4},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{3,4},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{3,5},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{4,5},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{2,4},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{2,5},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{2,3},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{2,3},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{2,5},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{2,4},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{4,5},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{3,5},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{3,4},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2,3,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{1},{2,3,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{1},{2,4,5},{3}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{1},{3,4,5},{2}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{2},{1,3,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{2},{1,3,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{2},{1,4,5},{3}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{3},{1,2,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{3},{1,2,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{4},{1,2,3},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000422
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[{1,2},{3}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[{1,3},{2}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[{2,3},{1}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[{1},{2,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{1},{2,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{1},{3,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{2},{1,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{2},{1,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{3},{1,2},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{4},{1,2},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{3},{1,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{4},{1,3},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{2},{3,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{3},{2,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{4},{2,3},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,2},{3},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,2},{4},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,3},{2},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,4},{2},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,3},{4},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,4},{3},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{2,3},{1},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{2,4},{1},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{3,4},{1},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{2,3},{4},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{2,4},{3},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{3,4},{2},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,2},{3,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,3},{2,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,4},{2,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 4
[{2,3},{1,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 4
[{2,4},{1,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 4
[{3,4},{1,2}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 4
[{1,2,3},{4}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[{1,2,4},{3}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[{1,3,4},{2}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[{2,3,4},{1}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 4
[{1},{2},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{2},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{2},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{3},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{3},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{4},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{5},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{4},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{5},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{3},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{4},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1},{5},{3,4},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{2},{1},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[{1,2},{3},{4},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,2},{3},{5},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,2},{4},{3},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,2},{5},{3},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,2},{4},{5},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,2},{5},{4},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,3},{2},{4},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,3},{2},{5},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,4},{2},{3},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,5},{2},{3},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,4},{2},{5},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,5},{2},{4},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,3},{4},{2},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,3},{5},{2},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,4},{3},{2},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,5},{3},{2},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,4},{5},{2},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,5},{4},{2},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,3},{4},{5},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,3},{5},{4},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,4},{3},{5},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,5},{3},{4},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,4},{5},{3},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{1,5},{4},{3},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,3},{1},{4},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,3},{1},{5},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,4},{1},{3},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,5},{1},{3},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,4},{1},{5},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,5},{1},{4},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,4},{1},{2},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,5},{1},{2},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{4,5},{1},{2},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,4},{1},{5},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,5},{1},{4},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{4,5},{1},{3},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,3},{4},{1},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,3},{5},{1},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,4},{3},{1},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,5},{3},{1},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,4},{5},{1},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,5},{4},{1},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,4},{2},{1},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,5},{2},{1},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{4,5},{2},{1},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,4},{5},{1},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{3,5},{4},{1},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{4,5},{3},{1},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,3},{4},{5},{1}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[{2,3},{5},{4},{1}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph Kn equals 2n−2. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 25%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 25%
Values
[{1,2},{3}] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 6
[{1,3},{2}] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 6
[{2,3},{1}] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 6
[{1},{2,3},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1},{2,4},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1},{3,4},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2},{1,3},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2},{1,4},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3},{1,2},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{4},{1,2},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3},{1,4},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{4},{1,3},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2},{3,4},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3},{2,4},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{4},{2,3},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2},{3},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2},{4},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,3},{2},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,4},{2},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,3},{4},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,4},{3},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,3},{1},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,4},{1},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3,4},{1},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,3},{4},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,4},{3},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3,4},{2},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2},{3,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,3},{2,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,4},{2,3}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,3},{1,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,4},{1,3}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{3,4},{1,2}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2,3},{4}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{1,2,4},{3}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{1,3,4},{2}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{2,3,4},{1}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{1},{2},{3,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{2},{3,5},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{2},{4,5},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{3},{2,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{3},{2,5},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{4},{2,3},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{5},{2,3},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{4},{2,5},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{5},{2,4},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{3},{4,5},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{4},{3,5},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{5},{3,4},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{2},{1},{3,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{2},{3},{4,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{3},{4,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{3},{5,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{4},{3,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{4},{3,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{5},{3,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{6},{3,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{5},{3,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{6},{3,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{4},{5,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{5},{4,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{2},{6},{4,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{2},{4,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{2},{4,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{2},{5,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{2},{3,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{2},{3,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{2},{3,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{2},{3,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{2},{3,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{2},{3,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{2},{5,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{2},{4,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{2},{4,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{4},{2,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{4},{2,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{5},{2,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{6},{2,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{5},{2,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{6},{2,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{3},{2,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{3},{2,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{3},{2,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{3},{2,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{3},{2,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{3},{2,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{5},{2,3},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{6},{2,3},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{4},{2,3},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{4},{2,3},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{6},{2,3},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{5},{2,3},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{5},{2,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{4},{6},{2,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{4},{2,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{4},{2,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{5},{6},{2,4},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{6},{5},{2,4},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{4},{5,6},{2}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
[{1},{3},{5},{4,6},{2}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 0 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000454
Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[{1,2},{3}] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 1
[{1,3},{2}] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 1
[{2,3},{1}] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 1
[{1},{2,3},{4}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1},{2,4},{3}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1},{3,4},{2}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2},{1,3},{4}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2},{1,4},{3}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{3},{1,2},{4}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{4},{1,2},{3}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{3},{1,4},{2}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{4},{1,3},{2}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2},{3,4},{1}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{3},{2,4},{1}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{4},{2,3},{1}] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,2},{3},{4}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,2},{4},{3}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,3},{2},{4}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,4},{2},{3}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,3},{4},{2}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,4},{3},{2}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2,3},{1},{4}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2,4},{1},{3}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{3,4},{1},{2}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2,3},{4},{1}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{2,4},{3},{1}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{3,4},{2},{1}] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,2},{3,4}] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,3},{2,4}] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,4},{2,3}] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[{2,3},{1,4}] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[{2,4},{1,3}] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[{3,4},{1,2}] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[{1,2,3},{4}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[{1,2,4},{3}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[{1,3,4},{2}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[{2,3,4},{1}] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{2},{3,5},{4}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{2},{4,5},{3}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{3},{2,4},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{3},{2,5},{4}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{4},{2,3},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{5},{2,3},{4}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{4},{2,5},{3}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{5},{2,4},{3}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{3},{4,5},{2}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{4},{3,5},{2}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1},{5},{3,4},{2}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{2},{1},{3,4},{5}] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[{1,2,3},{4},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,2,3},{5},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,2,4},{3},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,2,5},{3},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,2,4},{5},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,2,5},{4},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,3,4},{2},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,3,5},{2},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,4,5},{2},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,3,4},{5},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,3,5},{4},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,4,5},{3},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{2,3,4},{1},{5}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{2,3,5},{1},{4}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{2,4,5},{1},{3}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{3,4,5},{1},{2}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{2,3,4},{5},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{2,3,5},{4},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{2,4,5},{3},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{3,4,5},{2},{1}] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[{1,2,3,4},{5}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[{1,2,3,5},{4}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[{1,2,4,5},{3}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[{1,3,4,5},{2}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[{2,3,4,5},{1}] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[{1,2,3},{4},{5,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,3},{5},{4,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,3},{6},{4,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,4},{3},{5,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,5},{3},{4,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,6},{3},{4,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,4},{5},{3,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,4},{6},{3,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,5},{4},{3,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,6},{4},{3,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,5},{6},{3,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,2,6},{5},{3,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,4},{2},{5,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,5},{2},{4,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,6},{2},{4,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,4,5},{2},{3,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,4,6},{2},{3,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,5,6},{2},{3,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,4},{5},{2,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,4},{6},{2,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,5},{4},{2,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,6},{4},{2,5}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,5},{6},{2,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,3,6},{5},{2,4}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[{1,4,5},{3},{2,6}] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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